A waitress sold $13$ ribeye steak dinners and $26$ grilled salmon dinners, totaling $\$ 569.37$ on a particular day. Another day she sold $26$ ribeye steak dinners and $13$ grilled salmon dinners, totaling $\$ 582.68$ . How much did each type of dinner cost? $\newline$ The cost of ribeye steak dinners is $\$$ $\square$ and the cost of salmon dinners is $\$$ $\square$ . $\newline$ (Simplify your answer. Round to the nearest hundredth as needed.)

Full solution

Q. A waitress sold

13

ribeye steak dinners and

26

grilled salmon dinners, totaling

\$ 569.37

on a particular day. Another day she sold

26

ribeye steak dinners and

13

grilled salmon dinners, totaling

\$ 582.68

. How much did each type of dinner cost?

\newline

The cost of ribeye steak dinners is

\$

\square

and the cost of salmon dinners is

\$

\square

\newline

(Simplify your answer. Round to the nearest hundredth as needed.)

Formulate Equations: Let's call the cost of one ribeye steak dinner $R$ and the cost of one grilled salmon dinner $S$ . We have two equations based on the information given: $\newline$ $1$ st day: $13R + 26S = 569.37$ $\newline$ $2$ nd day: $26R + 13S = 582.68$
Multiply First Equation: Now, let's multiply the entire first equation by $2$ to make the coefficient of $R$ the same in both equations: $\newline$ $2\times(13R + 26S) = 2\times569.37$ $\newline$ $26R + 52S = 1138.74$
Eliminate R: Next, we'll subtract the second equation from this new equation to eliminate R: $\newline$ $(26R + 52S) - (26R + 13S) = 1138.74 - 582.68$ $\newline$ $39S = 556.06$
Find Cost of Salmon: Now, we divide both sides by $39$ to find the cost of one grilled salmon dinner: $\newline$ $S = \frac{556.06}{39}$ $\newline$ $S = 14.26$
Find Cost of Ribeye: We'll plug the value of $S$ back into the first equation to find $R$ : $\newline$ $13R + 26(14.26) = 569.37$ $\newline$ $13R + 370.76 = 569.37$ $\newline$ $13R = 569.37 - 370.76$ $\newline$ $13R = 198.61$
Find Cost of Ribeye: We'll plug the value of $S$ back into the first equation to find $R$ : $\newline$ $13R + 26(14.26) = 569.37$ $\newline$ $13R + 370.76 = 569.37$ $\newline$ $13R = 569.37 - 370.76$ $\newline$ $13R = 198.61$ Finally, we divide both sides by $13$ to find the cost of one ribeye steak dinner: $\newline$ $R = \frac{198.61}{13}$ $\newline$ $R = 15.28$